相关论文: Probable Counterexamples of the ABC Conjecture
The article provides a counterexample to a conjecture by Blocki-Zwonek.
We establish a lower bound for the size of possible counterexamples of the Dixmier Conjecture. We prove that $B>15$, where $B$ is the minimum of the greatest common divisor of the total degrees of $P$ and $Q$, where $(P,Q)$ runs over the…
By an $abc$ triple, we mean a triple $(a,b,c)$ of relatively prime positive integers $a,b,$ and $c$ such that $a+b=c$ and $\operatorname{rad}(abc)<c$, where $\operatorname{rad}(n)$ denotes the product of the distinct prime factors of $n$.…
We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy…
We propose a novel use of a recent new computational tool for Bayesian inference, namely the Approximate Bayesian Computation (ABC) methodology. ABC is a way to handle models for which the likelihood function may be intractable or even…
Presented here are over one hundred conjectures ranging from easy to difficult, from many mathematical fields. I also summarize briefly methods and tools that have led to this collection.
We provide a complete characterisation of the phenomenon of adversarial examples - inputs intentionally crafted to fool machine learning models. We aim to cover all the important concerns in this field of study: (1) the conjectures on the…
Counterfactual explanations (CEs) are methods for generating an alternative scenario that produces a different desirable outcome. For example, if a student is predicted to fail a course, then counterfactual explanations can provide the…
Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…
Counterfactual statements, e.g., "my headache would be gone had I taken an aspirin" are central to scientific discourse, and are formally interpreted as statements derived from "alternative worlds". However, since they invoke hypothetical…
We establish supercongruences for two kinds of Ap\'ery-like numbers, which involve Bernoulli numbers and Bernoulli polynomials. Conjectural supercongruences of the same type for another four kinds of Ap\'ery-like numbers are also proposed.
In this paper we study the problem of making predictions using multiple structural casual models defined by different agents, under the constraint that the prediction satisfies the criterion of counterfactual fairness. Relying on the…
This is a summary of the proof of BAB conjecture. All material are taken from the two BAB paper in the reference. The aim of this summary is to help reader to understand the more technical side of the proof of BAB.
We prove a conjecture posted in the Online Encyclopedia of Integer Sequences, namely that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3.…
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of…
Post-hoc interpretability approaches have been proven to be powerful tools to generate explanations for the predictions made by a trained black-box model. However, they create the risk of having explanations that are a result of some…
Approximate Bayesian Computation (ABC) has become increasingly prominent as a method for conducting parameter inference in a range of challenging statistical problems, most notably those characterized by an intractable likelihood function.…
Inference networks have a variety of important uses and are constructed by persons having quite different standpoints. Discussed in this paper are three different but complementary methods for generating and analyzing probabilistic…
We present new counterexamples, which provide stronger limitations to sums-differences statements than were previously known. The main idea is to consider non-uniform probability measures.
A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.